Correlation shaping matched filter receiver

ABSTRACT

The present invention is directed toward an apparatus and method for receiving signals, demodulating the signals and shaping the correlation of the output of a correlation demodulator. The present invention can be used irrespective of whether the received signals have noise components that are Gaussian or non-Gaussian. Moreover, the present invention can be utilized when a predetermined set of received signals is linearly independent or linearly dependent.

GOVERNMENT SUPPORT

[0001] This invention was made with government support under CooperativeAgreement DAAL01-96-2-0001 awarded by the U.S. Army Research Laboratory.The government has certain rights in the invention.

BACKGROUND OF THE INVENTION

[0002] Many of our present technologies use signal processing principlesto achieve their communication functions. In a typical signal detectionapplication, the receiver performs the function of receiving andprocessing signals. A design engineer using a receiver to perform signaldetection designs the receiver so that it is capable of receiving andprocessing a predetermined set of signals. These signals vary dependingupon the system in which the design engineer is deploying the receiver.Because signals received after transmission typically contain a noisecomponent, the receiver must process the received signal and extractinformation regarding the transmitted signal. The receiver may contain adetector or similar device, which could be used to determine whichsignal within the predetermined set of signals was in fact received bythe receiver.

[0003] Many prior art receivers perform signal detection by using acorrelation demodulator and a detector. One example of a correlationdemodulator is a matched filter demodulator. A single matched filter isbasically a linear filter whose transfer function has been matched to aparticular electronic signal or environment in order to perform afiltering that is optimum for some particular purpose. See generallyU.S. Pat. No. 4,044,241 “Adaptive Matched Filter,” the contents of whichare hereby incorporated by reference. The filter is particularly matchedto a signal plus noise input from which the signal is desired to beextracted. Id. A matched filter demodulator is comprised of a bank ofmatched filters the output of which is sampled at one time instant toobtain a vector signal.

[0004] The matched filter demodulator can equivalently be implemented asa correlation demodulator with correlating signals equal to thetime-reversed versions of the filters' transfer functions.

[0005] When one of a predetermined set of signals is received inadditive white Gaussian noise, the matched filter detector is optimalfor maximizing the probability of detection.

[0006] The matched filter detector is comprised of a correlationdemodulator, where the correlating signals are equal to thepredetermined set of signals that could have been received, followed bya detector. The detector declares as the detected signal the one forwhich the output of the correlator is a maximum. See generally J. G.Proakis, Digital Communications, McGraw-Hill, Inc. 3^(rd) ed 1995, thecontents of which are hereby incorporated by reference.

[0007] Although matched filter receivers are optimal when the addednoise is white and Gaussian, they are not when the received signalcontains added non-Gaussian noise, or other forms of distortion. Designengineers nonetheless continue to use matched filter receivers inoperating environments containing non-Gaussian or non-additive noise, inpart because optimal receivers for these noise environments aretypically nonlinear and, therefore, difficult to implement. Furthermore,in many applications the noise distribution is unknown to the receiver.See generally T. Kailath and V. Poor, Detection of Stochastic Processes,IEEE Transactions on Information Theory, vol. 44, pp. 2230-59, October1998, the contents of which are hereby incorporated by reference. Inlight of the fact that many receivers are required to perform insituations where the added noise is non-Gaussian or a combination ofGaussian and non-Gaussian, there is a need for a receiver which issimple to implement, does not rely on the channel parameters and canachieve an acceptable probability of detecting the correctly receivedsignal irrespective of whether the noise added to the received signal iswhite, Gaussian, or non-Gaussian.

SUMMARY OF THE INVENTION

[0008] According to one aspect, the present invention provides areceiver comprising a correlation demodulator and a correlation shaper.In one embodiment, the correlation shaper is a decorrelator. In anotherembodiment, the decorrelator is comprised of a whitening transformationperformed on a vector signal output from the correlation demodulator. Inyet another embodiment the correlation shaper outputs a vector signalwhose covariance matrix has the property that the second row and eachsubsequent row is a permutation of the first row. In another embodiment,a subspace whitening transformation is performed on a vector signaloutput from the correlation demodulator. In an additional embodiment,the correlation demodulator is comprised of a bank of correlators thatcross-correlate the received signal with a set of orthogonal signals. Inyet another embodiment, the correlation demodulator is comprised of abank of correlators that cross-correlate the received signal with a setof projected orthogonal signals. In an additional embodiment, thecorrelation demodulator is comprised of a bank of correlators thatcross-correlate the received signal with a set of geometrically uniformsignals. In yet another embodiment, the correlation demodulator iscomprised of a bank of correlators that cross-correlate the receivedsignal with a set of projected geometrically uniform signals. In stillanother embodiment, the correlation shape of a signal that is an outputof the correlation demodulator can be varied.

[0009] According to another aspect, the present invention provides amethod for processing signals comprising the steps of receiving a memberof a predetermined set of signals; demodulating the member of thepredetermined set of signals; and, shaping the correlation of thedemodulated signal.

BRIEF DESCRIPTION OF THE DRAWING

[0010] The invention is described with reference to the several figuresof the drawing, in which,

[0011]FIG. 1 is a block diagram of an illustrative receiver, which maybe used with the present invention.

[0012]FIG. 2 is a block diagram of an illustrative matched filterdemodulator, which may be used with the present invention.

[0013]FIG. 3 illustrates various embodiments of the receiver of thepresent invention.

[0014]FIG. 4 illustrates various embodiments of the receiver of thepresent invention.

[0015]FIG. 5 is a plot of test results for one embodiment of the presentinvention.

[0016]FIG. 6 is a plot of test results for one embodiment of the presentinvention.

[0017]FIG. 7 is a plot of test results for one embodiment of the presentinvention.

[0018]FIG. 8 is a plot of test results for one embodiment of the presentinvention.

[0019]FIG. 9 is a plot of test results for one embodiment of the presentinvention.

[0020]FIG. 10 is a plot of test results for one embodiment of thepresent invention.

[0021]FIG. 11 is a plot of test results for one embodiment of thepresent invention.

DETAILED DESCRIPTION

[0022] The present invention is directed toward a novel way of receivingand processing a distorted version of a signal, s_(k)(t), which is oneof a predetermined set of signals, {s₁(t), s₂(t) . . . s_(m)(t)} . Theone of the predetermined set of signals, s_(k)(t), lies in a realHilbert space H with inner product <x(t), y(t))>⟨x(t), y(t)⟩ = ∫_(t = −∞)^(∞)x(t)y(t)t.

[0023] The present invention is particularly useful in signal detectionapplications where the noise environment is not Gaussian.

[0024] A receiver is configured to receive a predetermined set ofsignals. This predetermined set of signals is comprised of a finitenumber of signals, and is hereinafter represented as {s_(k)(t), 1≦k≦m}.The actual signal received by the receiver, r(t), is comprised of onemember of the predetermined set of received signals, s_(k)(t), and anoise component, n(t). In order to perform its signal detectionfunction, a receiver must process the received signal prior to detectingwhich of the predetermined set of received signals it actually received.

[0025] The receiver and method of signal detection of the presentinvention allows an engineer to design a specific correlation shape forthe output of a correlation demodulator. In this way, the presentinvention overcomes the problems encountered with prior art matchedfilter processing methods, for example, by overcoming the fact that theoutput of a matched filter demodulator contains elements that arecorrelated among themselves.

[0026] In some of the embodiments provided herein, the design engineercould choose to have the output of the correlation demodulator becompletely decorrelated. In yet other embodiments the design engineercould choose to shape the correlation of the output of the correlationdemodulator such that the covariance matrix of this output has theproperty that the second row and each subsequent row is a permutation ofthe first row. In alternative embodiments, an engineer practicing thepresent invention can choose any correlation shape for the output of thecorrelation demodulator. By shaping the correlation of the output of thecorrelation demodulator, the design engineer can design a simplereceiver tailored to receive the predetermined set of signals.

[0027] The receiver and method of signal processing of the presentinvention are highly versatile and can be utilized irrespective ofwhether the predetermined set of received signals is linearly dependentor linearly independent. In addition, the received signal can bepre-whitened to facilitate signal processing when the received signalhas been corrupted by colored noise. Moreover, the invention disclosedherein can be implemented for both continuous-time signals and discretesignals. Embodiments disclosed herein may also be implemented inhardware, for example in DSP chips, or in software using C++ programminglanguage, for example. In addition to disclosing a receiver forprocessing signals, one aspect of the present invention also provides anovel method wherein a design engineer can alter the correlation shapeof the output of a correlation demodulator.

[0028] A basic overview of the present invention is illustrated inFIG. 1. With reference now to FIG. 1, a signal r(t) 20 is first receivedby a receiver. The signal is then processed by a correlation demodulator30 which may provide some correlation shaping of the vector signaloutput. The vector signal output of the correlation demodulator may thenbe additionally shaped by a correlation shaper 40. The output of thecorrelation shaper could be passed to a detector or similar device. Thedetector or similar device could process the output and determine whichmember of a predetermined set of signals was received by the receiver.The detector or similar device could output signal, s_(k)(t) 60 in FIG.1.

[0029] The embodiments are generally directed toward differentimplementations of the present invention. In some embodiments, a matchedfilter demodulator may be used. In these embodiments, the correlation ofthe output of the matched filter demodulator is shaped using a lineartransformation. In other embodiments, the shaping of the correlation ofthe demodulator output is done within the demodulator by choosing adifferent set of signals than those used to carry out thecross-correlation in the matched filter demodulator. These embodimentsof the present invention are physically distinct, but mathematicallyequivalent. This equivalence is shown in the following section. In thenotation that follows, the symbol W is used with reference to awhitening transformation. One skilled in the art will appreciate thatthese equations are generally applicable to the case where thetransformation is not a whitening transformation.

[0030] Mathematical Equivalence of Embodiments

[0031] A. Problem Formulation

[0032] Suppose we have a transmitter that transmits one of M signals{s_(k)(t), 1≦m≦M} with equal probability, where the signals lie in areal Hilbert space H with inner product <x(t),y(t))>=∫x(t)y(t)dt. Weassume only for simplicity of exposition here that the signals arelinearly independent and normalized so that ∫s_(m) ²(t)dt=1 for all m.For a discussion of the more general case of linearly dependent signalssee Y. C. Eldar, A. V. Oppenheim, and D. Egnor, “Orthogonal andProjected Orthogonal Matched Filter Detection,” submitted to IEEE Trans.on Signal Proc. January 2001. The received signal r(t) is modeled asr(t)=s_(k)(t)+n(t), where n(t) is a stationary white noise process withzero mean and spectral density².

[0033] We demodulate the signal r(t) using a correlation demodulator asdepicted in FIG. 4. The received signal r(t) is cross-correlated with Mnormalized signals q_(m)(t)εH so that a_(m)=(q_(m)(t), r(t)), where thesignals q_(m)(t) are to be determined. When the signals q_(m)(t) areorthogonal the receiver is referred to as an orthogonal matched filterreceiver. The detected signal is s_(k)(t) where k=argmax a_(m). Thedifference between the orthogonal matched filter (“OMF”) receiver andthe matched filter (“MF”) receiver lies in the choice of the signalsq_(m)(t).

[0034] If the transmitted signal is s_(k)(t), then

a _(m) =<q _(m)(t),r(t))=<q _(m)(t),s_(k)(t))+<q _(m)(t),n(t))  (1)

[0035] The detected signal will be the transmitted signal s_(k)(t) ifmax_(m) |q_(m)(t), s_(k)(t)+n(t))=((t), s_(k)(t)+n(t)). Therefore, wewould like to choose the signals q_(m)(t) to maximize (q_(m)(t),s_(m)(t)) for 1≦m≦M. It is well known that the signals q_(m)(t)=s_(m)(t)maximize this inner product. The resulting demodulator is thenequivalent to the well known MF demodulator. We note thatq_(m)(t)=s_(m)(t) also maximizes the sum${R_{hs} = {\sum\limits_{m = 1}^{M}\quad {\langle{{q_{m}(t)},{s_{m}(t)}}\rangle}}},$

[0036] since the individual terms are maximized by this choice. We willm=l see shortly that when additional constraints are imposed, it will beuseful to consider maximizing the sum rather than the individual terms.

[0037] In general, the outputs am of the demodulator are correlatedsince they share information regarding the noise process n(t). We wouldlike to choose the signals q_(m)(t) so that the outputs a_(m) areuncorrelated. When the noise is non-Gaussian this approach does in factlead to improved performance over conventional MF detection in manycases.

[0038] Let cov (a_(m), a_(k)) denote the cross-covariance of a_(m) anda_(k). Then,

cov(a _(m) , a _(k))=E(<q _(m)(t),n(t))(n(t),q _(k)(t)>)=σ² <q(t),q_(k)(t)>.  (2)

[0039] From (2) it follows that the outputs of the demodulator areuncorrelated if and only if the signals q_(m)(t) are orthonormal, i.e.,if and only if <q_(m)(t), q_(k)(t))=δ_(mk) for all m, k. We thereforepropose to choose the signals q_(m)(t) to be orthonormal.

[0040] As before, we would also like to choose the signals q_(m)(t) tomaximize <q_(m)(t), s_(m)(t)) for 1≦m≦M. However, we now have anadditional constraint, namely that the signals q_(m)(t) are orthonormal.If the signals s_(m)(t) are not orthonormal, then we cannot maximize theinner products individually subject to this constraint. Instead, weconsider maximizing the sum of the inner products. Thus, we seek a setof signals {q_(m)(t), 1≦m≦M} such that $\begin{matrix}{R_{hs} = {\sum\limits_{m = 1}^{M}\quad {\langle{{q_{m}(t)},{s_{m}(t)}}\rangle}}} & (3)\end{matrix}$

[0041] is maximized, subject to the constraint

<q _(m)(t)q _(k)(t)>=δ_(mk), 1≦m≦M.  (4)

[0042] The design problems of (3) and (4) can be formulated in twoequivalent ways to provide further insight into the problem.Specifically, it will be shown that the following problems are the same:

[0043] 1. Find a set of orthonormal signals {q_(m)(t}, 1≦m ≦M} thatmaximize R_(hs)=Σ_(m)<s_(m)(t),q_(m)(t));

[0044] 2. Find an optimal whitening transformation W that minimizes thetotal mean squared error (MSE) between the whitened output b=Wã and theinput ã, where ã denotes the vector output of the conventional MFdemodulator. Then choose the signals {q_(m)(t), 1 m M} to be theorthonormal signals given by q_(m)(t)=Σ_(k)W*_(mk)s_(k)(t); and,

[0045] 3. Find a set of orthonormal signals {q_(m)(t), 1 m M} that areclosest in a least-squares sense to the signals {s_(m)(t), 1 m M},namely that minimize ε=Σ_(m)<s_(m)(t)−q_(m)(t), s_(m)(t)−q_(m)(t)).

[0046] Turning now to showing the equivalence between the three problemsabove, let M S:C^(M)→H denote the linear transformation defined by${{Sx} = {\sum\limits_{m = 1}^{M}\quad {x_{m}{s_{m}(t)}}}},$

[0047] where xεC^(M) is an arbitrary M-dimensional vector and x_(m)denotes the m-th component of x. Let S*:H<C^(M) denote the adjointtransformation so that if x=S*y(t) for arbitrary y(t)εH, thenx_(m)=<s_(m)(t)y(t)). Let ã denote the vector with the m-th componentã_(m), where ã_(m) is the output of the correlation demodulator whenq_(m)(t)=s_(m)(t). From (2) it follows that the covariance matrix of ãdenoted C_(a), is given by

C _(a) =σ ² S*S,  (5)

[0048] where the mk-th element of S*S is <s_(m)(t), s_(k)(t)>. If thesignals s_(m)(t) are not orthonormal, then C_(a) is not diagonal and theelements of ã are correlated. Suppose we whiten ã using a whiteningtransformation W to obtain the random vector b=Wã, where the covariancematrix of b is given by C_(b)=σ²I, and then base our detection on b.Thus, the components b_(m) are the inputs to the detector, and thedetected signal is s_(k)(t) if k=argmaxb_(m). Since the detector basesits decision on the vector b, we choose a whitening transformation Wthat minimizes the MSE given by $\begin{matrix}{{ɛ_{mse} = {\sum\limits_{m = 1}^{M}\quad {E\left( \left( {b_{m}^{\prime} - {\overset{\sim}{a}}_{m}^{\prime}} \right)^{2} \right)}}},{{{where}\quad {\overset{\sim}{a}}_{m}^{\prime}} = {{{\overset{\sim}{a}}_{m} - {{E\left( \overset{\sim}{a_{m}} \right)}\quad {and}\quad b_{m}^{\prime}}} = {b_{m} - {{E\left( b_{m} \right)}.}}}}} & (6)\end{matrix}$

[0049] That is, from all possible whitening transformations, we seek theone that results in a white vector b as close as possible to theoriginal vector ã.

[0050] We now show that the demodulator depicted in FIG. 3 is equivalentto the correlation demodulator of FIG. 4 where the signals q_(m)(t) areorthonormal and given by q_(m)(t)Σ_(k)=k W_(mk)*S_(k)(t) where W_(mk)denotes the mk th element of W. In other words, the outputs of FIG. 3, 4are equal, provided that q_(m)(t)=Σ_(k)W_(mk)*s_(k)(t)

[0051] The vector output b of FIG. 3 is given by

b=Wã=WS*r(t)=Q*r(t)  (7)

[0052] where Q: C^(M)→H is given by Q=SW*. Therefore, b can be viewed asthe output of a correlation demodulator with signalsq_(m)(t)=Σ_(k)W_(mk)*s_(k)(t).

[0053] We now need to show that the signals q_(m)(t) are orthonormal. Itis sufficient to show that Q*Q=WS*SW*=I. By definition, C_(b)=σ²I. Inaddition, C_(b)=WC_(a)W* and from (5) C_(a)=σ²S*S. Therefore,${Q^{*}Q} = {{{WS}^{*}{SW}^{*}} = {{\frac{1}{\sigma^{2}}C_{b}} = {I.}}}$

[0054] In summary, the output of FIG. 3 may be obtained using thecorrelation demodulator of FIG. 4, where the signals q_(m)(t) areorthonormal and given by q_(m)(t)=Σ_(k)W_(mk)*s_(k)(^(t))

[0055] We now show the minimization of ε_(mse) given by (6), isequivalent to maximization of R_(hs) given by (3). Using (7) we have

b−ã=(Q*−S*)r(t)=(Q*−S*)(s _(k)(t)+n(t))  (8) and $\begin{matrix}{{b_{m}^{\prime} - {\overset{\sim}{a}}_{m}^{\prime}} = {{\langle{{{q_{m}(t)} - {s_{m}(t)}},{n(t)}}\rangle}.}} & (9)\end{matrix}$

[0056] Substituting (9) into (6) we have $\begin{matrix}{{ɛ_{mse} = {{\sum\limits_{m = 1}^{M}{{{E\left( {\langle{{{q_{m}(t)} - {s_{m}(t)}},{n(t)}}\rangle}^{2} \right)}.{Let}}\quad e_{m}}} = {{E\left( {\langle{{{s_{m}(t)} - {q_{m}(t)}},{n(t)}}\rangle}^{2} \right)}.\quad {Then}}}},\quad {e_{m} = {E\left( {\int\limits_{t}{\left( {{s_{m}(t)} - {q_{m}(t)}} \right){n(t)}{t}}} \right)}^{2}}} & (10)\end{matrix}$

$\begin{matrix}{= {{\int_{t,t^{1}}{\left( {{s_{m}(t)} - {q_{m}(t)}} \right)\left( {{s_{m}\left( t^{\prime} \right)} - {q_{m}\left( t^{\prime} \right)}} \right){E\left( {{n(t)}{n\left( t^{\prime} \right)}} \right)}\quad {t}{t^{\prime}}}} = {{\sigma^{2}{\int{\left( {{s_{m}(t)} - {q_{m}(t)}} \right)^{2}{t}}}} = {\sigma^{2}{\langle{{{s_{m}(t)} - {q_{m}(t)}},{{s_{m}(t)} - {q_{m}(t)}}}\rangle}}}}} & (11)\end{matrix}$

[0057] Combining (10) and (11) we see that minimization of ε_(mse) isequivalent to minimization of ε_(is), where $\begin{matrix}{ɛ_{ls} = {\sum\limits_{m = 1}^{M}{{\langle{{{s_{m}(t)} - {q_{m}(t)}},{{s_{m}(t)} - {q_{m}(t)}}}\rangle}.}}} & (12)\end{matrix}$

[0058] Therefore, the optimal whitening problem is equivalent to theproblem of finding a set of orthonormal signals {q_(m)(t), 1≦m≦M} thatare closest in the least-squares sense to the signals {s_(m)(t), 1≦m≦M}.

[0059] Finally, we show that this least-squares problem is equivalent toour original design problem of (3) and (4). Expanding ε_(is), we have$\begin{matrix}\begin{matrix}{{ɛ_{ls} =}\quad} \\{= {\sum\limits_{m = 1}^{M}\left( \quad {{\langle{{s_{m}(t)},{s_{m}(t)}}\rangle} + {\langle{{q_{m}(t)},{q_{m}(t)}}\rangle} - {2{\langle{{s_{m}(t)},{q_{m}(t)}}\rangle}}} \right)}} \\{= {\sum\limits_{m = 1}^{M}\left( {2 - {2{\langle{{s_{m}(t)},{q_{m}(t)}}\rangle}}} \right)}}\end{matrix} & (13)\end{matrix}$

[0060] From (3) and (13) it follows that minimization of ε_(ls) isequivalent to maximization of R_(hs). Since minimization of ε_(mse) isequivalent to minimization of ε_(ls), we conclude that these threeproblems are equivalent.

[0061] Note, that if the transmitted signals s_(m)(t) are orthonormal,then the output of the MF demodulator ã is white. Thus, in this case W=Iand the OMF detector is equivalent to the MF detector. Alternatively, ifthe signals s_(m)(t) are orthonormal, then the residual least-squareserror ε_(ls) is minimized when q_(m)(t)=s_(m)(t), and again the OMFdetector reduces to the MF detector.

[0062] B. Optimal Whitening

[0063] Since the optimal whitening problem is equivalent to the problemof (3)-(4), we can choose to determine the signals {q_(m)(t), 1≦m≦M} bysolving this problem.

[0064] We first restate the optimal whitening problem in its mostgeneral form. Let aεR^(M) be a random vector with m-th component a_(m)and positive-definite covariance matrix Ca, and leta′_(m)=a_(m)−E(a_(m)). We seek a whitening transformation W such thatthe white vector b =Wa has a covariance matrix C_(b)=σ²I, and is asclose as possible to a in the MSE sense. Thus, we seek thetransformation W that minimizes $\begin{matrix}{ɛ_{mse} = {\sum\limits_{m = 1}^{M}{E\left( \left( {a_{m}^{\prime} - b_{m}^{\prime}} \right)^{2} \right)}}} & (14)\end{matrix}$

[0065] where b_(m) is the mth component of b, and b′_(m)=b_(m)−E(b_(m)),subject to the constraint

C _(b) =WC _(a) W*=σ ² I  (15)

[0066] where C_(b) is the covariance matrix of b. Since W must beinvertible (15) reduces to

σ²(W*W)⁻¹ =C _(a)  (16).

[0067] We solve this minimization problem using the eigendecompositionof C_(a) and the singular value decomposition (SVD) of W.

[0068] Let the vectors ν_(k) denote the orthonormal eigenvectors ofC_(a), so that

C _(a)ν_(k)=λ_(k)ν_(k),1≦k≦M  (17)

[0069] where λ_(K)>0. We can then decompose C_(a) as C_(a)=VDV* where Vdenotes the unitary matrix of columns ν_(k) and D denotes the diagonalmatrix with diagonal elements λ_(K). Then

W*Wν _(k)=σ² C _(a) ⁻¹ν_(k)=σ²λ_(k) ⁻¹ν_(k)  (18).

[0070] From the properties of the SVD, it then follows that

Wν_(k)=σ_(k) u _(k),1≦k≦M  (19)

[0071] where $\sigma_{k} = {\sigma/\sqrt{\lambda_{k}}}$

[0072] and the vectors u_(k) are orthonormal.

[0073] Since the M vectors ν_(k)εC^(M) are orthonormal, they span thespace C^(M) and any xεC^(M) may be expressed as x=Σ_(k)<ν_(k), x>ν_(k)where the inner product on C^(M) is defined as <ν_(k), x>=ν_(k)*x. Leta′=a−E(a) and b′=b−E(b). Then a′=Σ_(k)<ν_(k),a′>ν_(k) and

b′−a′=Wa′−a′=Σ_(k) , a′)(σ_(k) u _(k) −ν _(k)),  (20)

[0074] where we used (19). We can now express ε_(mse) of (14) as$\begin{matrix}{ɛ_{mse} = {{E\left( {\langle{{b^{\prime} - a^{\prime}},{b^{\prime} - a^{\prime}}}\rangle}^{2} \right)} = {\sum\limits_{k,m}{{E\left( {{\langle{v_{m},a^{\prime}}\rangle}{\langle{a^{\prime},v_{k}}\rangle}} \right)}{\langle{{{\sigma_{k}u_{k}} - v_{k}},{{\sigma_{m}u_{m}} - v_{m}}}\rangle}}}}} & (21)\end{matrix}$

[0075] Now,

E(<ν_(m) , a′> <a′,ν _(k))=<ν_(m) ,C _(a)ν_(k)>=λ_(k)<ν_(m),ν_(k)>=λ_(k)δ_(mk)  (22)

[0076] Substituting (22) in (21) results in $\begin{matrix}{{ɛ_{mse} = {{\sum\limits_{k}{\lambda_{k}{\langle{{{\sigma_{k}u_{k}} - v_{k}},{{\sigma_{k}u_{k}} - v_{k}}}\rangle}}} = {\sum\limits_{k}{\lambda_{k}\left( {\sigma_{k}^{2} + 1 - {2\sigma_{k}{\Re \left( {\langle{u_{k},v_{k}}\rangle} \right)}}} \right)}}}},} & (23)\end{matrix}$

[0077] where

(•) denotes the real part. From (23) it follows that minimizing ε_(mse)is equivalent to maximizing A=Σ_(k)σ_(k)

(<u_(k), ν_(k)>). Using the Cauchy-Schwartz inequality

[0078] we have,

A=Σ_(k)σ_(k)

(<u _(k),ν_(k)>)  (24) $\begin{matrix}{\leq {\sum\limits_{k}{\sigma_{k}{{\langle{u_{k},v_{k}}\rangle}}}}} & (25) \\{{\leq {\sum\limits_{k}{\sigma_{k}\left( {{\langle{u_{k},u_{k}}\rangle}{\langle{v_{k},v_{k}}\rangle}} \right)}^{1/2}}} = {\sum\limits_{k}\sigma_{k}}} & (26)\end{matrix}$

[0079] with equality in (25) if and only if <u_(k), ν_(k)> is real andnonnegative, and equality in (26) if and only if u_(k)=c_(k)ν_(k) forsome nonzero constants c_(k). Since the vectors u_(k) are orthonormal<u_(k),u_(k)=1. We therefore conclude that A≦Σ_(k)σ_(k) with equality ifand only if u_(k)=ν_(k). Thus, ε_(mse) is minimized when W is given by$\begin{matrix}{{Wv}_{k} = {{\sigma_{k}v_{k}} = {\frac{\sigma}{\sqrt{\lambda_{k}}}v_{k}}}} & (27)\end{matrix}$

[0080] or

W=σVD ^(−½) =V*=σC _(a) ^(−½)  (28)

[0081] In summary, the optimal whitening transformation that minimizesthe MSE ε_(mse) defined in (14) for an input a with covariance C_(a) andan output b with covariance C_(b)=σ²I, is W=σC_(a) ^({fraction (1/2)}.)

[0082] In FIG. 3 the input to the whitening transformation is a=ã withC_(a)=σ²S*S. Thus, the optimal whitening transformation in this case isW=(S*S)^(−½), and the optimal orthonormal signals q_(m)(t) that maximizeR_(hs) are given by q_(m)(t)=Σ_(k)W*_(km)s_(k)(^(t)) or q_(m)(t) =Qi_(m)where Q=S(S*S)^(−½) and i_(m)(k)=δ_(mk). If the signals s_(k)(t) arelinearly dependent, then W=((S*S)^(½))†where ( )† denotes theMoore-Penrose pseudo-inverse.

[0083] C. Covariance Matrix of the Output is Arbitrary:

[0084] The correlation shaper in FIG. 3 can be chosen so that thecovariance matrix is arbitrary within the mathematical constraintsimposed on any covariance matrix. In this case the correlation shapercan be chosen so that WC_(a)W*=C_(b).

[0085] D. Covariance Matrix of the Output has the Permutation Property:

[0086] Suppose we choose the correlation shaper in FIG. 3 so that thecovariance matrix of the output has the property that the second row andeach subsequent row is a permutation of the first row.

[0087] Let d_(k) be the elements of the first row of the specifiedcovariance matrix. The correlation shaper that minimizes the MSE betweenthe input and output is given as follows.

[0088] Let D be a diagonal matrix whose diagonal elements are thesquare-roots of the generalized Fourier transform of the sequence d_(k);the generalized Fourier transform is defined on a group formed by theelements of the pre-specified covariance matrix. See generally, Y. C.Eldar, G. D. Forney, Jr., “On quantum detection and the square-rootmeasurement”, IEEE Trans. on Inform. Theory, vol. 47, No. 3, March 2001,the contents of which are hereby incorporated by reference.

[0089] Let F be a Fourier matrix representing the generalized Fouriertransform over the group formed by the elements of the covariancematrix.

[0090] For linearly independent signals:

W=SFD(DF*S*SFD)^(−½) DF*

[0091] For linearly dependent signals:

W=SFD((DF*S*SFD)^(−½)†) DF*

[0092] 1. Orthogonal/Projected Orthogonal & Geometrically Uniform &Projected Geometrically Uniform Correlating Signals

[0093] In all cases the closest signals (in a least-squares sense) tothe predetermined set of received signals are given by the equation${q_{k}(t)} = {\sum\limits_{l = 1}^{m}\quad {{S_{l}(t)}(W)_{lk}}}$

[0094] where (W)_(lk) is the lk-th element of W. As was previously thecase, one skilled in the art will recognize that the these equations areequally valid when the transformation is not a whitening transformation.

[0095] As previously mentioned, in some of the embodiments, thecorrelation demodulator 30 is a matched filter demodulator. Referencewill now be made to FIG. 2. The matched filter demodulator 130 receivesan incoming signal, r(t) 120, and processes the received signal 120through a plurality of multipliers 132 and integrators 134. Theplurality of multipliers 132 and integrators cross-correlate thereceived signal 120 with the predetermined set of signals {s₁(t),s₂ (t). . . s_(m)(t)}. After this cross-correlation, the matched filterdemodulator 130 outputs a correlated signal {ã₁, ã₂ . . . ãm} 136.

[0096] An illustration of a preferred embodiment of the presentinvention is provided in FIG. 3. FIG. 3 illustrates a received signalr(t) 120, the matched filter demodulator 130 of FIG. 2, demodulatoroutput signal {ã₁, ã₂ . . . ã_(m)} 136, a correlation shaper 140, and anoutput of the correlation shaper {b₁, b₂ . . . b_(m)} 142. Fourembodiments will be discussed with reference to FIG. 3. The embodimentsvary depending on the correlation shape chosen by the design engineerand upon whether the predetermined set of signals {s₁(t), s₂(t) . . .s_(m)(t)} is linearly independent or linearly dependent.

[0097] Embodiment No. 1—Linearly Independent Received Signals &Decorrelated Output

[0098] In the first of these four embodiments, it is assumed that thecorrelation shape chosen is to have the output {b₁, b₂ . . . b_(m)} becompletely decorrelated. In addition, in this embodiment, thepredetermined set of signals {s₁(t), s₂(t) . . . s_(m)(t)} is linearlyindependent. In this embodiment, the correlation shaper 140 performs awhitening transformation on the matched filter demodulator output signal136. After the whitening transformation, the output of the correlationshaper 140, which was correlated when it emerged from the matched filterdemodulator 130, becomes uncorrelated. In this embodiment, it may bedesirable to minimize the mean squared error between the output of thecorrelation shaper 142 and the output of the matched filter demodulator136. This embodiment may perform satisfactorily for a given system evenif the covariance shaper does not result in the smallest mean squarederror value between the output of the correlation shaper 142 and thematched filter demodulator 136.

[0099] Embodiment No. 2—Linearly Independent Received Signals & Outputwith Specific Correlation Shape

[0100] An alternative embodiment of the present invention, also shown inFIG. 3, allows a design engineer to specify a correlation shape for theoutput of the correlation shaper 140, depicted as {b₁, b₂ . . . b_(m)}in FIG. 3. The correlation shape of {b₁, b₂ . . . b_(m)} is given by thecovariance matrix of b, 142, where b is a vector with components b_(k).An engineer can alter the correlation shape of b 142 by choosing thecovariance matrix to have specific properties. In addition, one skilledin the art may decide in certain circumstances to allow the correlationshape of the output of the correlation shaper to be arbitrary. In thisinstance, the covariance matrix can be comprised of arbitrary valuesthat satisfy the constraints imposed on any covariance matrix.

[0101] This embodiment differs from the one previously described, inwhich the correlation shaper decorrelated its output by performing awhitening transformation on the input of the correlation shaper. Rather,in this embodiment, the transformation is chosen so that the output ofthe correlation shaper has an arbitrary correlation shape.

[0102] In this embodiment, it may additionally be desirable to minimizethe mean squared error between the output of the correlation shaper 142and the output of the matched filter demodulator 136. We assume that thesecond and each subsequent row is chosen to be a permutation of thefirst. An engineer may alter the correlation shape of the output byaltering the permutations of the rows of the covariance matrix.

[0103] This embodiment may perform satisfactorily for a given systemeven if the correlation shaper does not result in the smallest meansquared error value between the output of the correlation shaper 142 andthe matched filter demodulator 136.

[0104] Embodiment No. 3—Linearly Dependent Received Signals &Decorrelated Output

[0105] In an alternative embodiment of the inventive receiver, theoutput of the correlation shaper 140 can be decorrelated when thepredetermined set of signals {s₁(t), s₂(t) . . . s_(m)(t)} is linearlydependent. When the predetermined set of signals is linearly dependent,the outputs of the matched filter demodulator are deterministicallylinearly dependent, meaning that vector ã and the elements of b=Wã arealso linearly dependent. The linear dependence of the predetermined setof signals renders conventional whitening techniques impossible. Thus,in this alternative embodiment, the outputs of the matched filterdemodulator will be whitened on the subspace upon which they lie.

[0106] Making reference now to FIG. 3, the outputs of the matched filterdemodulator 136 lie in a subspace V of R^(m). In this embodiment, it isdesirable to whiten the outputs of the matched filter demodulator {ã₁,ã₂, . . . ã_(m)} 136 on the subspace V, hereinafter referred to assubspace whitening. In this embodiment, subspace whitening is defined sothat the whitened signal lies in V and its representation in terms ofany orthonormal basis for V is white.

[0107] In this embodiment, it may be desirable to minimize the meansquared error between the output of the correlation shaper 142 and theoutput of the matched filter demodulator 136. This embodiment mayperform satisfactorily for a given system even if the correlation shaperdoes not result in the smallest mean squared error value between theoutput of the correlation shaper 142 and the matched filter demodulator136.

[0108] Embodiment No. 4—Linearly Dependent Received Signals & Output HasSpecific Correlation Shape An additional embodiment of the presentinvention can be used when the predetermined set of signals {s₁(t),s₂(t) . . . s_(m)(t)} is linearly dependent. As opposed to EmbodimentNo. 3 discussed above, an engineer utilizing this embodiment wouldchoose a correlation shape for the output of the correlation shaper 140.In this embodiment, the correlation shaping performed by the correlationshaper 140 is similar to that described in Embodiment No. 2, except thatcorrelation shaping in this embodiment is performed on a subspacespanned by the predetermined set of signals {s₁(t),s₂(t) . . . s_(m)(t)}. In this embodiment, it may desirable to minimize the mean squarederror between the output of the correlation shaper 142 and the output ofthe matched filter demodulator 136. This embodiment may performsatisfactorily for a given system even if the correlation shaper doesnot result in the smallest mean squared error value between the outputof the correlation shaper 142 and the matched filter demodulator 136.

[0109] Embodiment No. 5-8—Choosing a Set of Orthogonal Signals

[0110] An alternative embodiment of the present invention is depicted inFIG. 4. The receiver 210 comprises a received signal r(t) 220, acorrelation demodulator 230, a set of output signals {a₁, a₂ . . .a_(m)} from the correlation demodulator 240, and one of a predeterminedset of signals received by the receiver, s_(k)(t) 260. In thisembodiment, the correlation shaping is achieved by selecting the signals{q₁(t), q₂(t) . . . q_(m)(t)} 232. The selection of this set of signalsvaries depending upon the desired correlation shape of the output vector{a₁, a₂ . . . a} 240 and upon whether the predetermined set of signals{s₁(t), s₂(t) . . . s_(m)(t)} is linearly independent or linearlydependent. Alternative embodiments for each of these variations arediscussed below.

[0111] Embodiment No. 5—Linearly Independent Received Signals &Orthogonal Signals Making reference now to FIG. 4, the first of theseembodiments will be discussed. The output signal {a₁, a₂. . . a_(m)} 240is decorrelated and the predetermined set of signals {s₁(t), s₂(t) . . .s_(m)(t)} is linearly independent. If the desired correlation shape isto have the output of the correlation demodulator 230 be decorrelated,the set of signals {q₁(t), q₂(t). . . q_(m)(t)} 233 should beorthogonal. In this embodiment, it may be desirable to minimize theleast squares error between the orthogonal signals {q₁(t), q₂(t) . . .q_(m)(t)} 233 and the predetermined set of signals {s₁(t), s₂(t) . . .s_(m)(t)}. While minimizing this least-squares value is preferred inthis embodiment, those skilled in the art will recognize that adequateperformance may be achieved although the least-squares error may not beat its minimum.

[0112] Embodiment No. 6—Linearly Independent Received Signals &Geometrically Uniform Signals

[0113] If an engineer desired a particular correlation shape for {a₁, a₂. . . a_(m)} 240, the engineer could achieve this result by altering thechoice of the signal set {q₁(t), q₂(t) . . . q_(m)(t)} 233. If thecovariance of the output of the correlation shaper has the property thatthe second and each subsequent row is chosen to be a permutation of thefirst, this can be achieved by choosing {q₁(t), q₂(t) . . . q_(m)(t)} tobe geometrically uniform. See generally G. D. Forney, Jr.,“Geometrically Uniform Codes,” IEEE Trans. Inform. Theory, vol. IT-37,No. 5, pp. 1241-60, Sep. 1991, the contents of which are herebyincorporated by reference. It may be desirable in this embodiment tominimize the least- squares error between the geometrically uniformsignals and the predetermined set of signals {s₁(t), s₂(t) . . .s_(m)(t)}. Thus, an alternative embodiment of the present inventioncomprises minimizing the least-squares error between {q₁(t), q₂(t) . . .q_(m)(t)} and {s₁(t), s₂ (t) . . . s_(m) (t)} and requiring {q₁(t),q₂(t) . . . q_(m)(t)} to be geometrically uniform.

[0114] Embodiment No. 7—Linearly Dependent Signals & ProjectedOrthogonal Signals

[0115] If the desired correlation shape is to have the output of theoutput of the correlation demodulator be decorrelated on the space inwhich it lies, the set of signals {q₁(t), q₂(t) . . . q_(m)(t)} shouldbe chosen as a projected orthogonal signal set. In yet anotherembodiment to be used when the predetermined set of signals is linearlydependent, the signals, {q₁(t), q₂(t) . . . q_(m)(t)} 233 in FIG. 4, arethe closest projected orthogonal signals to the predetermined set ofsignals, {s₁(t), s₂(t) . . . s_(m)(t)}, in a least-squares sense. Theprojected orthogonal signals are a projection of a set of orthonormalsignals onto the space spanned by the predetermined set of signals,{s₁(t), s₂(t) . . s_(m)(t)}. Analogously to the previous embodiment,skilled artisans will recognize that, depending upon particular systemconstraints, an absolute minimum least-squares value need not beattained in order to achieve adequate performance.

[0116] Embodiment No. 8—Linearly Dependent Received Signals & ProjectedGeometrically Uniform Signals

[0117] The present invention can also be used to achieve a particularcorrelation shape among the members of the output vector a 240 when thepredetermined signals {s₁(t), s₂(t) . . . s_(m)(t)} are linearlydependent. If the covariance matrix of a 240 when represented in termsof an orthonormal basis for the space in which it lies has the propertythat subsequent rows are permutations of the first, then {q₁(t), q₂(t) .. . q_(m)(t)} should be selected to be a set of projected geometricallyuniform set of signals, i.e., a set of geometrically uniform signalsprojected onto the space spanned by the predetermined set of signals. Inan alternative embodiment, the signals {q₁(t), q₂(t) . . . q_(m)(t)} 233are chosen to be the closest projected geometrically uniform signals ina least-squares sense to the signals {s₁(t), s₂(t) . . . s_(m)(t)}. Inthis way, an engineer is able to determine a correlation shape for {a₁,a₂ . . . a_(m)} 240, when the predetermined set of signals {s₁(t), s₂(t). . . s_(m)(t)} is linearly dependent.

[0118] Embodiment No. 9—Method of Performing Signal Detection

[0119] An additional embodiment comprises a method for processing asignal from a predetermined set of signals, wherein the method includesthe steps of receiving a member of a predetermined set of signals;demodulating the member; outputting a demodulated signal; and shapingthe correlation of the demodulated signal.

[0120] Test Results for Embodiment Nos. 5 & 7

[0121] The results of tests conducted using an implementation ofEmbodiment Nos. 5 & 7 are provided below. In each test, signal detectionwas executed and the results compared to the signal detectioncapabilities of a matched filter. One skilled in the art will recognizethat the signal detection properties tested herein are indicative of theperformance that the present invention would exhibit when implemented aspart of a receiver or as part of a method of signal detection. Two typesof random signals were used, each 100 samples in length: (1) binarysignals where each sample takes on the value 0 or 1 with equalprobability; and, (2) computer simulated analog signals where eachsample is uniformly distributed between 0 and 1. The non-Gaussian noisedistribution was chosen as a Gaussian mixture of two components withequal weights.

[0122] For ease of notation, the embodiments of FIG. 4 will be referredto in short hand. In the case where the predetermined set of receivedsignals is linearly independent, the orthogonal matched filter will bereferred to as “OMF.” In the case of where the predetermined set ofreceived signals is linearly dependent, the projected orthogonal matchedfilter will be referred to as “POMF.” Both the OMF and the POMF weretested in Gaussian and non-Gaussian noise environments.

[0123] A Gaussian mixture of two components each with standard deviation0.25 centered at ±1, corresponding to a signal-to-noise-ratio (“SNR”)close to 0 dB was tested first. Five hundred realizations of each typeof signal were generated. For each realization of the signal, aprobability of detection was determined by recording the number ofsuccessful detections over 500 noise realizations. The histograms of theprobability of detection P_(d) for the different detectors were plotted.These histograms indicated that P_(d) has a unimodal distribution with abell-shaped appearance. The results of these tests are presented interms of the mean and standard deviation of P_(d) for the variousdetectors in FIG. 5 and FIG. 6.

[0124] Reference will now be made to FIG. 5 and FIG. 6. These figurescontain the mean of P_(d) for the OMF detector and the matched filterdetector as a function of the number of signals in the transmittedconstellation. The vertical lines indicate the standard deviation ofP_(d). The results in FIG. 5 were obtained for binary signals, and theresults in FIG. 6 for computer simulated analog signals. Both figuresdemonstrate the superior results obtained by the OMF detector whencompared with the matched filter detector.

[0125]FIG. 7 is a plot of the mean and standard deviation of P_(d) forthe POMF detector and the matched filter detector as a function of thenumber of signals in the transmitted constellation. In this simulation,the Gaussian mixture of two components had a standard deviation 0.25centered at ±1 . For each signal constellation, ⅔ (rounded up) of thesignals were linearly independent. The dependent signals were generatedby randomly mixing a set of linearly independent signals where thesignal and mixing elements were uniformly distributed between 0 and 1.

[0126] The qualitative behavior of the POMF detector is similar to thatof the OMF detector, even when compared under various Gaussian mixtureparameters and SNRs. FIGS. 5-7 demonstrate the superior results obtainedwith OMF and POMF detectors when compared to the matched filter detectorin a non-Gaussian noise environment.

[0127] These tests were repeated in a Gaussian noise environment, theresults of which are provided in FIGS. 8-10. The details of thesimulations were identical to those of FIGS. 5, 6, and 7, respectively,with the Gaussian noise replacing the Gaussian mixture noise. FIGS.11(a) and (b) are plots of the difference in the probability ofdetection (on a scale of 10⁻³) using the OMF and matched filterdetectors for transmitted constellations of 15 signals in Gaussiannoise, for binary and computer simulated analog signals respectively.FIG. 11(c) is a plot of the difference in the probability of detectionusing the POMF and matched filter detectors in Gaussian noise. Thequalitative behavior of the OMF and POMF detectors in comparison to thematched filter detector is similar for varying values of the variance ofthe Gaussian noise.

[0128] For Gaussian noise the matched filter detector outperformed theOMF and the POMF detectors. This appears consistent with the fact thatthe matched filter detector maximizes the probability of detection forGaussian noise. However, it is evident from FIGS. 8-11 that the relativeimprovement in performance using the matched filter detector over theOMF and POMF detectors is insignificant.

[0129] Other embodiments of the invention will be apparent to thoseskilled in the art from a consideration of the specification or practiceof the invention disclosed herein. It is intended that the specificationand examples are illustrative only. Although various embodiments havebeen described in detail, it should be understood that various changes,substitutions, and alterations can be made herein by one of ordinaryskill in the art without departing from the scope of the presentinvention as hereinbefore described and as hereinafter claimed. What isclaimed is:

1. A receiver comprising: a correlation demodulator for receivingsignals; and a correlation shaper operating on an output from thecorrelation demodulator.
 2. The receiver of claim 1 wherein thecorrelation demodulator is a matched filter demodulator.
 3. The receiverof claim 1, wherein the correlation shaper is a whitening transformationperformed on the output from the correlation demodulator.
 4. Thereceiver of claim 3, wherein the whitening transformation is determinedby minimizing the mean squared error between the output from thecorrelation demodulator and the output from the correlation shaper. 5.The receiver of claim 1, wherein the correlation shaper is comprised ofa transformation, said transformation is determined by minimizing themean squared error between the output of the correlation demodulator andthe output of the correlation shaper.
 6. The receiver of claim 1,wherein the correlation shaper is chosen so that the covariance matrixof the output has the property that the second and subsequent rows arepermutations of the first row.
 7. The receiver of claim 6, wherein thecorrelation shaper is determined by minimizing the mean squared errorbetween the output from the correlation demodulator and the output fromthe correlation shaper.
 8. The receiver of claim 1, wherein thecorrelation shaper is a subspace whitening transformation performed onthe output signal of the correlation demodulator.
 9. The receiver ofclaim 8, wherein the subspace whitening transformation is determined byminimizing the mean squared error between the output from thecorrelation demodulator and the output from the correlation shaper. 10.The receiver of claim 5, wherein the transformation is performed on asubspace.
 11. The receiver of claim 1, wherein the correlation shaper ischosen so that the covariance matrix of the representation of the outputof the correlation shaper in the space in which it lies has the propertythat the second and subsequent rows are permutations of the first row.12. The receiver of claim 11, wherein the correlation shaper isdetermined by minimizing the mean squared error between the output fromthe correlation demodulator and the output from the correlation shaper.13. The receiver of claim 1, wherein the correlation demodulator iscomprised of a bank of correlators that cross-correlate a receivedsignal with a set of orthogonal signals.
 14. The receiver of claim 13,wherein the set of orthogonal signals is determined by minimizing theleast-squares error between the set of orthogonal signals and thepredetermined set of signals.
 15. The receiver of claim 1, wherein thecorrelation demodulator is comprised of a bank of correlators thatcross-correlate a received signal with a set of geometrically uniformsignals.
 16. The receiver of claim 15, wherein the set of geometricallyuniform signals is determined by minimizing the least-squares errorbetween the set of geometrically uniform signals and the predeterminedset of signals.
 17. The receiver of claim 1, wherein the correlationdemodulator is comprised of a bank of correlators that cross-correlate areceived signal with a set of projected orthogonal signals.
 18. Thereceiver of claim 17, wherein the set of projected orthogonal signals isdetermined by minimizing the least-squares error between the set ofprojected orthogonal signals and the predetermined set of signals. 19.The receiver of claim 1, wherein the correlation demodulator iscomprised of a bank of correlators that cross-correlate a receivedsignal with a set of projected geometrically uniform signals.
 20. Thereceiver of claim 19, wherein the set of projected geometrically uniformsignals is determined by minimizing the least-squares error between theset of projected geometrically uniform signals and the predetermined setof signals.
 21. A method for processing signals comprising the steps of:receiving a member of a predetermined set of signals that has undergonesome distortion; demodulating the received signal of the predeterminedset of signals; outputting a demodulated signal; and shaping thecorrelation of the demodulated signal.
 22. The method of claim 21,wherein the shaping of the correlation of the demodulated signal iscomprised of performing a whitening transformation on the demodulatedsignal.
 23. The method of claim 22, wherein the whitening transformationis comprised of the step of minimizing the mean squared error betweenthe demodulated signal and an output signal from the whiteningtransformation.
 24. The method of claim 21, wherein shaping of thecorrelation of the demodulated signal is further comprised of performinga transformation on the demodulated signal, said transformation isdetermined by minimizing the mean squared error between the demodulatedsignal and the output of the correlation shaper.
 25. The method of claim21, wherein shaping of the correlation of the demodulated signal iscomprised of performing a transformation of the demodulated signal suchthat the covariance matrix of the output of the transformation has theproperty that the second and each subsequent row is a permutation of thefirst.
 26. The method of claim 25, wherein the transformation iscomprised of the step of minimizing the mean squared error between thedemodulated signal and an output signal from the transformation.
 27. Themethod of claim 21, wherein shaping of the correlation of thedemodulated signal is comprised of performing a subspace whiteningtransformation on the demodulated signal.
 28. The method of claim 27,wherein the whitening transformation is comprised of the step ofminimizing a mean squared error between the demodulated signal and anoutput signal from the whitening transformation.
 29. The method of claim21, wherein shaping of the correlation of the demodulated signal iscomprised of performing a transformation of the demodulated signal suchthat the covariance matrix of the representation of the output of thetransformation on the space in which it lies has the property that thesecond and each subsequent row is a permutation of the first.
 30. Themethod of claim 29, wherein the transformation is comprised of the stepof minimizing a mean squared error between the demodulated signal and anoutput signal from the transformation.
 31. The method of claim 21,wherein whitening the demodulated signal is comprised of the step ofcross-correlating the member of a predetermined set of signals with aset of orthogonal signals.
 32. The method of claim 31, furthercomprising the step of minimizing the least- squares error between thepredetermined set of signals and the set of orthogonal signals.
 33. Themethod of claim 21, wherein shaping the correlation of the demodulatedsignal is comprised of the step of cross-correlating the member of apredetermined set of signals with a set of geometrically uniformsignals.
 34. The method of claim 33, further comprising the step ofminimizing the least- squares error between the predetermined set ofsignals and the set of geometrically uniform signals.
 35. The method ofclaim 21, wherein shaping the correlation of the demodulated signal on asubspace is comprised of the step of cross-correlating the member of apredetermined set of signals with a set of projected orthogonal signals.36. The method of claim 35, comprising the step of minimizing theleast-squares error between the projected orthogonal signals and thepredetermined set of signals.
 37. The method of claim 21, whereinshaping the correlation of the demodulated signal on a subspace iscomprised of the step of cross-correlating the member of a predeterminedset of signals with a set of projected geometrically uniform signals.38. The method of claim 37, comprising the step of minimizing theleast-squares error between the projected geometrically uniform signalsand the predetermined set of signals.